Mathematical Sciences: Weighted Norm Inequalities and Partial Differential Equations

数学科学：加权范数不等式和偏微分方程

• 批准号: 9003095
• 负责人: Cristian Gutierrez
• 金额: $ 3.87万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-05-01 至 1992-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003095&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Weighted; Norm; Inequalities

This project is concerned with problems in two subareas of mathematical analysis: partial differential equations and harmonic analysis related to the Gaussian measure. The project has two objectives. The first is to analyze the behavior of solutions of a class of degenerate parabolic equations in divergence form with non-smooth coefficients. Such coefficients may be zero, infinite or both. The equations are known to model cases of diffusion of temperature in a non-homogeneous and non-isotropic material. Basic questions to be considered in this work include finding conditions which imply regularity of solutions, determining whether a Harnack principle is valid for non-negative solutions and understanding the behavior of the fundamental solutions of the equations. Weighted norm inequalities of the Poincare and Sobolev type will be used in this study. The second objective concerns the behavior, in the space of functions integrable with respect to Gaussian measure, of a class of transforms generated by the Ornstein-Uhlenbeck semigroup. These transforms play the role, in the Gaussian context, of the classical singular integrals of M. Riesz. Of particular interest will be the establishment of weak-type estimates with constants bounded independently of dimension. This investigation will center on the use of a maximal function as in the Calderon- Zygmund theory, but this function does not inherit all the properties of the classical maximal functions. However, there is evidence to suggest that it is still of weak type with respect to Gaussian measure. This is the first problem to be resolved.

这个项目涉及数学分析的两个子领域的问题：偏微分方程和与高斯测度相关的谐波分析。该项目有两个目标。首先分析了一类具有非光滑系数的散度型退化抛物方程解的性质。这些系数可以是零、无穷大或两者兼而有之。已知这些方程用于模拟温度在非均匀和非各向同性材料中的扩散情况。在这项工作中要考虑的基本问题包括寻找暗示解的正则性的条件，确定哈纳克原理是否对非负解有效，以及理解方程基本解的行为。本研究将使用庞加莱和索博列夫型的加权范数不等式。第二个目标是研究由Ornstein-Uhlenbeck半群生成的一类变换在高斯测度可积函数空间中的行为。在高斯背景下，这些变换扮演了M. Riesz经典奇异积分的角色。特别令人感兴趣的将是建立具有独立于维数的有界常数的弱型估计。本研究将集中在Calderon- Zygmund理论中极大函数的使用上，但是这个函数并没有继承经典极大函数的所有性质。然而，有证据表明，它仍然是弱类型相对于高斯测度。这是第一个要解决的问题。